Calculus is a branch of Mathematics concerned with 'rates of change'. In the 17th century two mathematicians, Newton and Leibniz, developed similar ideas independently, each using their own notations. Calculus has two major sections : Differentiation and Integration.

Differentiation is the process of finding the derivative of a function at a given point.

Derivative is the ratio of the change in the value of the function to change in the independent variable. It measures the steepness of the graph of the function at a point on the graph, thus, it is the slope.

Tangent to a Curve

Applications

A Stationary Point is where the graph is not increasing or decreasing but flat. At stationary points, the gradient of the graph is zero as the graph momentarily is parallel to the x-axis.

Often (but not always) these stationary points are also turning points, if the graph turns around and heads in he other direction. A Turning Point is when the graph stops increasing (or decreasing) and starts decreasing (or increasing)

Kinematics is the study of movement. Displacement is the distance in a particular direction. 

• The rate of change of displacement is velocity.

• The rate of change of velocity is acceleration.

• If we have an expression for displacement and we differentiate it, we get an expression for velocity.

• If we have an expression for velocity and we differentiate it, we get an expression for acceleration.

Anti- differentiation or integration is the inverse process of differentiation.  The symbol ∫ is used to denote the integral and dx (which comes from Leibniz notation) tells us which variable we are finding the anti-derivative of.